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Taylor-Socolar hexagonal tilings as model sets

The Taylor-Socolar tilings are regular hexagonal tilings of the plane but are distinguished in being comprised of hexagons of two colors in an aperiodic way. We place the Taylor-Socolar tilings into an algebraic setting which allows one to see them directly as model sets and to understand the corresponding tiling hull along with its generic and singular parts. Although the tilings were originally obtained by matching rules and by substitution, our approach sets the tilings into the framework of a cut and project scheme and studies how the tilings relate to the corresponding internal space. The centers of the entire set of tiles of one tiling form a lattice $Q$ in the plane. If $X_Q$ denotes the set of all Taylor-Socolar tilings with centers on $Q$ then $X_Q$ forms a natural hull under the standard local topology of hulls and is a dynamical system for the action of $Q$. The $Q$-adic completion $\bar{Q}$ of $Q$ is a natural factor of $X_Q$ and the natural mapping $X_Q \longrightarrow \bar{Q}$ is bijective except at a dense set of points of measure 0 in $\bar{Q}$. We show that $X_Q$ consists of three LI classes under translation. Two of these LI classes are very small, namely countable $Q$-orbits in $X_Q$. The other is a minimal dynamical system which maps surjectively to $\bar{Q}$ and which is variously $2:1$, $6:1$, and $12:1$ at the singular points. We further develop the formula of Socolar and Taylor (2011) that determines the parity of the tiles of a tiling in terms of the co-ordinates of its tile centers. Finally we show that the hull of the parity tilings can be identified with the hull $X_Q$; more precisely the two hulls are mutually locally derivable.